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The Helly bound for singular sums

2002
*
Discrete Mathematics
*

In this paper we will establish a

doi:10.1016/s0012-365x(01)00240-0
fatcat:r4iq5hjkcnbr5o7tn2unrs4vgm
*bound**for**the*number of*sums*which can be generated by a clique in*the**singularity*graph of Zn,*the*ring of integers modulo n. ... When n has at least three prime factors, there are always cliques based on*Helly*families of sets which realize n − (n)*sums*, where denotes*the*Euler totient function. ...*The*author is grateful to*the*referees*for*suggesting that it could be extended. ...##
###
PROOF OF A CONJECTURE OF STEINHAUS

1954
*
Proceedings of the National Academy of Sciences of the United States of America
*

By

doi:10.1073/pnas.40.3.205
pmid:16589456
pmcid:PMC527972
fatcat:u4ubvcjq5vgxnfmbfs4nd4vrfy
*the**Helly*theorem there is a function ,(x) of*bounded*variation and a subsequence of j such that ... Steinhaus has conjectured'*the*following theorem: Suppose a trigonometric series -CD is given with*the*property that*the*partial*sums*N E aneinx -N are non-negative*for*all x and all sufficiently large ...##
###
On a topological fractional Helly theorem
[article]

2005
*
arXiv
*
pre-print

We prove a new fractional

arXiv:math/0506399v1
fatcat:zedt4cghpbcphnnasoc26mbsvy
*Helly*theorem*for*families of sets obeying topological conditions. ... This implies fractional*Helly*number k+1*for*families F. Moreover, we obtain a topological (p,q)-theorem. ... Acknowledgement I thank Carsten Schultz*for*suggesting*the*use of*the*spectral sequence argument, and my advisor Günter M. Ziegler*for*helpful discussions. Many thanks also to Jiří ...##
###
Bounding Helly Numbers via Betti Numbers
[chapter]

2017
*
A Journey Through Discrete Mathematics
*

Here βi denotes

doi:10.1007/978-3-319-44479-6_17
fatcat:7grpm6zbwrcvtbdpfkbwxawj7q
*the*reduced Z 2 -Betti numbers (with*singular*homology). ... If F is a finite family of subsets of R d such that βi ( G) ≤ b*for*any G F and every 0 ≤ i ≤ d/2 − 1 then F has*Helly*number at most h(b, d). ... We would like to express our immense gratitude to Jiří Matoušek, not only*for*raising*the*problem addressed in*the*present paper and valuable discussions about it, but, much more generally,*for**the*privilege ...##
###
Page 239 of American Mathematical Society. Proceedings of the American Mathematical Society Vol. 6, Issue 2
[page]

1955
*
American Mathematical Society. Proceedings of the American Mathematical Society
*

,> bice-t-
k=—oo As p tends to infinity through

*the*sequence of values pi, pz, - - -*the**sum*tends to d, = , bic g_x. k=0 But*the*total variation of*the*measures is uniformly*bounded*, and so by*the**Helly*...*The*first lemma asserts directly that yp is*singular*. LemMA.*The**singular*measure yp is also absolutely continuous. If K is a constant larger than any | },|, then & @ ld.] ...##
###
Bounding Helly numbers via Betti numbers
[article]

2016
*
arXiv
*
pre-print

Here β̃_i denotes

arXiv:1310.4613v3
fatcat:xdowys3xjbe7jjta3yxaeddjsq
*the*reduced Z_2-Betti numbers (with*singular*homology). ... We show that very weak topological assumptions are enough to ensure*the*existence of a*Helly*-type theorem. ... ,*for**the*privilege of having known him, as our teacher, mentor, collaborator, and friend. ...##
###
Author index

2002
*
Discrete Mathematics
*

.,

doi:10.1016/s0012-365x(02)00277-7
fatcat:s5wqhobycrgn3kc2st32twlh4m
*The**Helly**bound**for**singular**sums*(1-3) 117-133 Faudree, R.J., R.J. Gould, M.S. Jacobson and L.L. ... ., Hall's condition*for*list-coloring, and*the*Hall parameters: recent developments (1-3) 135-147 K .undgen, A., Minimum average distance subsets in*the*hamming cube ...##
###
Page 247 of Mathematical Reviews Vol. 4, Issue 9
[page]

1943
*
Mathematical Reviews
*

[MF 7334]
Let S be a partially ordered vector space in which

*the*existence of an upper*bound**for*a monotone increasing sequence implies*the*existence of a least upper*bound*. ... It is shown that w(Z) compactness of order X, implies*the*extended*Helly*property of order X,, and that, if*the*extended*Helly*property holds*for*all orders, then*the*space is w(Z) complete. ...##
###
Bounding Radon's number via Betti numbers
[article]

2019
*
arXiv
*
pre-print

Using

arXiv:1908.01677v2
fatcat:vtjoq4rdo5g7xb7oe3i6ik6on4
*the*recent result of*the*author and Kalai, we manage to prove*the*following optimal*bound*on fractional*Helly*number*for*families of open sets in a surface: Let F be a finite family of open sets ... Then if F is a finite family of sets in X such that β_i( G; Z_2) is at most b*for*all i=0,1,..., k and G⊆ F, then*the*Radon's number of F is*bounded*in terms of b and X. ... Finally, many thanks to Natan Rubin*for*several discussions at*the*very beginning of*the*project. ...##
###
Quantitative combinatorial geometry for concave functions
[article]

2020
*
arXiv
*
pre-print

Our results also

arXiv:1908.04438v2
fatcat:22g7iyfoeffmjkl3zoyzh6riri
*bound**the*complexity of finding*the*best approximation of a family of convex sets by a single zonotope or by a single H-convex set. ... Our results characterize conditions that are sufficient*for**the*intersection of a family of convex sets to contain a "witness set" which is large under some concave or log-concave measure. ...*The*authors would like to thank Emo Welzl*for*his helpful comments and*for*pointing out*the*algorithmic applications of our results in*the*framework of LP-type problems. ...##
###
Development of Hankel Singular-Hypergraph Feature Extraction Technique for Acoustic Partial Discharge Pattern Classification

2021
*
Energies
*

Recent research posits that features extracted by

doi:10.3390/en14061564
fatcat:7vdaandqyzdizg4seuqjukfn4a
*singular*value decomposition (SVD) can exhibit*the*natural characteristics and energy contained in*the*signal. ...*The*algorithm is tested*for*various measurement conditions that include*the*influences of various PD locations and oil temperatures. ... Acknowledgments:*The*authors thank*the*Management of SASTRA University and Tata Realty-IT City-SASTRA Srinivasa Ramanujan Research cell. ...##
###
On a Theorem of Szego

1955
*
Proceedings of the American Mathematical Society
*

Jfc-«-00 As p tends to infinity through

doi:10.2307/2032347
fatcat:euch3o55fbfq7moknjdw7g54ri
*the*sequence of values pi, p2, • • •*the**sum*tends to 00 dq = £ bkCq-k. fc-0 But*the*total variation of*the*measures is uniformly*bounded*, and so by*the**Helly*theorem ... Since p, is a*singular*measure,*for*almost every point x this variation is o(e) as e tends to zero. ...##
###
On a theorem of Szegö

1955
*
Proceedings of the American Mathematical Society
*

Jfc-«-00 As p tends to infinity through

doi:10.1090/s0002-9939-1955-0074504-0
fatcat:4oupg5k6enczdgjirzqzh4elsi
*the*sequence of values pi, p2, • • •*the**sum*tends to 00 dq = £ bkCq-k. fc-0 But*the*total variation of*the*measures is uniformly*bounded*, and so by*the**Helly*theorem ... Since p, is a*singular*measure,*for*almost every point x this variation is o(e) as e tends to zero. ...##
###
Bounding Radon Number via Betti Numbers

2020
*
International Symposium on Computational Geometry
*

Using

doi:10.4230/lipics.socg.2020.61
dblp:conf/compgeom/Patakova20
fatcat:tehtzvsm6rc7jip64hcbut3fqu
*the*recent result of*the*author and Kalai, we manage to prove*the*following optimal*bound*on fractional*Helly*number*for*families of open sets in a surface: Let F be a finite family of open sets ... to k, then*the*Radon number of F is*bounded*in terms of b and X. ... From these consequences only*the*fact that*for*sets in R d*bounded*T C d/2 implies*bounded**Helly*number has been shown earlier [7] . ...##
###
Helly numbers of acyclic families

2014
*
Advances in Mathematics
*

In this paper, we prove topological

doi:10.1016/j.aim.2013.11.004
fatcat:3yns3opur5ghzkqspopjix32t4
*Helly*-type theorems*for*families of non-connected sets, that is, we give upper*bounds*on*Helly*numbers*for*such families. ... As an application, we obtain several explicit*bounds*on*Helly*numbers in geometric transversal theory*for*which only ad hoc geometric proofs were previously known; in certain cases,*the**bound*we obtain ...*The*authors would like to thank Jürgen Eckhoff*for*helpful comments on a preliminary version of this paper. ...
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